Probabilistic shaping for arbitrary modulation formats

Systems and methods for optical data transport, including controlling data transport across an optical transmission medium by generating two-dimensional (2D) distribution matchers (DMs) based on probabilistic fold shaping (PFS) and arbitrary probabilistic shaping (APS). The 2D PFS-based DM is can encode any N-fold rotationally symmetrical Quadrature Amplitude Modulation (QAM) format by applying the 2D PFS-based DM only to symbols in one quadrant based on a target entropy. A fold index yield uniform distribution is determined, and is utilized to carry generated uniform distributed parity check bits across the optical transmission medium. The 2D APS-based DM can encode any arbitrary modulation formats by encoding uniform binary data to generate non-uniform target symbols, and generating a probability distribution for the target symbols by indirectly applying the 2D APS-based DM based on a target probability distribution and a detected code rate of generated FEC code.